Self-Parameterized Chaotic Map for Low-Cost Robust Chaos
Round 1
Reviewer 1 Report
The paper introduces a general method with leads to the change of a system parameter of a chaotic map at every iteration and the widening of the chaotic parameter region.
1) The authors classify the chaotic systems in Section 1. They distinguish between 1D maps and multi-dimensional maps, and state that they limit their discussion to 1D maps. However, the introduced method is nothing else than the introduction of another variable instead of the system parameter C, so the examined system becomes two-dimensional. The dynamics of the new variable is given by another mapping, but the maps of xn and C are coupled. Thus, I think that the comparison of the original 1D maps and the new 2D maps is not quite fair: it is not surprising that the 2D maps outperform the 1D maps.
2) The authors determine the design parameters M and B based on inequalities (2)-(4). According to their reasoning, the fixed point regions and period-2 regions can be avoided by this approach. I am not convinced. The 2D map (xn,Cn) -> (xn+1,Cn+1) is expected to have periodic solutions with periodically varying C values. Moreover, the higher periods (3 or more) are not examined. Actually, traces of periodic solutions can be seen in Fig. 10.
3) It is unclear to me why M and B are written in boldface in Eq. (1). This notation suggests that M and B are vectors or matrices, however, both are scalars. Eq. (2) gives two inequalities that correspond to (3) and (4). I suggest either writing down the two inequalities of (2) separately, or joining (3) and (4) just like in (2).
4) On page 5 row 175, the authors refer to the x-axis in Figure 6. However, there is no x-axis there, only B and M axes.
5) The slopes of the seed maps are depicted in Figures 4 and 5 at the intersection between the transfer curve and the diagonal line. Actually, there are two intersections. Which one is considered here?
6) I don't understand how the transfer curves of the SPM maps can be interpreted. The given values of C are initial values, not parameters. I suspect that the 2D attractor resides at high values of C on the (x,C) plane. Perhaps this attractor could be depicted in a figure.
7) I suggest writing down the full names of the maps, coefficients and dimensions in the figure captions, instead of the abbreviations like SM, SPM, CC, CD.
8) How was the embedding dimension e= 2 determined in Section 3.2.3? This value can be satisfactory in case of a 1D map, but the SPM maps are 2D. According to 'Nonlinear Time Series Analysis' by H Kantz and T Schreiber (Cambridge University Press), the embedding dimension should be larger than 2D, where D is the box-counting (fractal) dimension.
9) According to the Conclusion, '...widening the chaotic range is demonstrated in detail with the help of the stability analysis of non-linear dynamics.' If the authors refer to the determination of the slopes at the intersection points: this is a method of _linear_ stability analysis, and the chaotic nature of the SPM maps is not proven by the corresponding results.
10) There is a possibility to make chaotic maps more irregular by a method that is not mentioned in the literature review. As it is shown in
DIGITAL CONTROL AS SOURCE OF CHAOTIC BEHAVIOR INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 20(5), pp. 1365-1378, DOI:10.1142/S0218127410026538, digital effects (i.e., sampling and quantization) may turn even linear maps to chaotic ones. I suggest mentioning this paper in the literature review. I think that this contribution may provide the authors with new ideas to improve the performance of the considered maps.Author Response
The authors would like to thank the reviewer for the valuable comments. We have thoroughly addressed the comments in our response attached.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper presents a self-parameterization method to provide wider chaotic regions in 1-D chaotic map designs. Overall, the paper is well-written and easy to follow. The design is technically sound and described in detail. The experiments are comprehensively designed. An FPGA-based implementation makes the design promising. A minor issue of the paper is that the experimental results do not highlight the benefits of the proposed design against prior arts very much.
Author Response
The authors would like to thank the reviewer for the valuable comments. We have thoroughly addressed the comments in our response attached.
Author Response File: Author Response.pdf
Reviewer 3 Report
This paper presents a general method called “self-parameterization" for designing one dimensional (1-D) chaotic maps that provide wider chaotic regions compared to existing 1-D maps. The ideas and methods in this paper have some novelty and has reference significance for chaotic system modeling. However, there are the following points need to be considered and revised by the author.
1. First of all, in the introduction, authors should clearly highlight the novelty and contribution of the work, preferably with a bulleted or enumerated list.
2. The definition of robust chaos needs to be supported by references.
3. In what aspects does the new chaotic map reflect robust chaos?
4. The comparison of approximate entropy(AE) between the new map and the existing maps should be given.
5. For the randomness test of the chaotic sequence generated by the new chaotic map, NIST test results should be given.
6. Acknowledgments appear in the front of the Conclusion section is wrong typesetting.
Author Response
The authors would like to thank the reviewer for the valuable comments. We have thoroughly addressed the comments in our response attached.
Author Response File: Author Response.pdf
Reviewer 4 Report
The paper reports a method that authors call "self-parametrization" for designing 1D chaotic maps. The proposed approach is of interest to scholars in the field and can find plenty of applications in chaos-based cryptography and communications systems. 1. "Self-parameterization" in the abstract: possible typo. 2. The authors claim that chaotic behavior can be observed in "nonlinear deterministic dynamic systems". What about stochastic chaos then? Can chaotic behavior be observed in the systems, described by SDEs? 3. The "classification" of chaotic systems into "two groups ... depending on the number of state variables involved" is at least questionable. First, this classification is different for continuous and discrete chaotic systems. Second, it is unclear how non-autonomous systems are treated in this classification. Third, what about fractional-order systems? Fourth, to the knowledge of the authors, impulse systems exist as well, which are quantized only in the time domain. I recommend rewriting this paragraph and completely revising this classification, or at least, providing the necessary references and theoretical grounds for it. 4. "in this work we limit our discussion to 1D discrete-time maps" - please, clarify this choice. It is known, that 1D maps suffer from low keyspace issues and are easy to break using various identification techniques in security applications. Moreover, the 1D maps are prone to dynamical degradation. 5. I recommend considering in the literature review recent techniques, connected with discrete chaotic maps modification, e.g. 1D coupled hyperbolic tangent chaotic map with delay, one dimensional piecewise chaotic map, discrete chaotic maps obtained using symmetric integration, and adaptive chaotic maps in cryptography applications I also recommend paying less attention to the general history of chaos theory: e.g. Lorenz's studies are mentioned twice, while they were not targeted to one-dimensional maps. 6. The authors systematically confuse analog and digital, software and hardware systems. For example, FPGA implementation of the chaotic map does not differ much from MCU implementation and is significantly different from the CMOS circuit. Besides, the simulation of the CMOS circuit, performed in some SPICE \ CAD system, is a numerical simulation (i.e. digital!) as well. Therefore, a question arises: how do the Authors compare these different digital implementations? How can the performance of a real analog chaotic generator be evaluated using digital simulation, which is prone to chaos degradation due to discretization effects? 7. The common procedure to evaluate chaotic sequences is statistical testing, e.g. using the NIST package. I recommend expanding the experimental section by adding NIST tests. 8. As far as this reviewer see, there were no real analog devices created during this study, and only simulations were used. Please, evaluate the possible complexities which appear while transiting from simulated CMOS circuits to real ones. 9. Please, reduce using the word "Moreover" in the introduction section. It is a bit tautological now. 10. The Authors claim that their approach suits well for hardware-constrained applications. However, these applications are not discussed in the Conclusions section and the hardware costs of the proposed solution are not evaluated quantitatively. Nevertheless, I like the study and believe it can be accepted for publication after moderate revisions.
Author Response
The authors would like to thank the reviewer for the valuable comments. We have thoroughly addressed the comments in our response attached.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
I accept the authors' responses. Indeed, I misunderstood the concept in my first review, now it is made clear.
Reviewer 3 Report
In my opinion, the authors have taken into account the observations and comments made previously. Therefore, I recommend to accept this manuscript.
Reviewer 4 Report
Thank you for providing the revised version of your manuscript and the very thorough reply letter. I am impressed by the amount of revisions done. I especially value the experiments with the NIST tests added. I have no more questions and can warmly recommend your paper for publication.
